Interaction between 3 Beams

With three beams, symmetric alignment and asymmetric alignment can be discussed. When aligned in symmetric configuration, we mean that the side beams below and above are identical in all dimensions including its shape, size, and also the distance between the middle beam. Ideally, the response is expected to be similar to that of the two beams, but with a doubled k coefficient. Two

voltage peaking frequencies are expected. If three voltage peaking frequencies are desired, asymmetric design can be considered: the side beams should be set in different distances to the main beam. To reduce complexity, identical beam parameters are used for the two side beams.

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Figure 4-6 Interactions between three beams (a) MCK model (b) schematic (c) block diagram

Similarly, the three-beam configuration is as shown in Figure 4-6 (b). The MCK models of the three beams is plotted in Figure 4-6 (a). Figure 4-6 (c) shows the block diagram for setting up the PSIM – Matlab simulation. By tuning the distances D12 and D13, we aim to create two additional peaks beside the resonance frequency of the middle beam, close enough to merge the three together.

Figure 4-7 shows the Power SIM simulation configurations.

Figure 4-7 PSIM Simulation diagram for 3 beams

4.2.2.1 The Symmetric Alignment

To create the symmetric alignment, the two side beams are set with identical parameters. Then, we set D12 and D13 in Figure 4-7 to be identical so that the system is symmetric. The middle beam is designed to have a different resonance frequency, which is lower than that of the two. It was predicted that, the optimal effect would appear when only two targeting resonance frequency f0 and 2f0 exist.

By the simulation, we can visualize the optimal distance between the beams and how the resonance frequency is influenced by the magnetic force.

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Figure 4-8 Simulation results for the symmetric 3-Beam alignment

(a) 3D plot of the middle beam (b) Frequency-Distance plot of middle beam (c) 3D plot of the side beams (d) Voltage-Frequency plot on D12=0.072m separate and combined

From the 3D plot Figure 4-8 (a), one can observe that there is an additional voltage peak similar to that of the two-beam alignment. However, it is noteworthy that there exists an optimal distance which the influence is the highest. To compare, we look to the 3D plot showing the output of the side beams (Figure 4-8 (c)), there is a small valley on its resonance, where we could clearly inference it from the damping when interacting with the middle beam - that the energy is transferred to the middle beam. Another difference can be observed in Figure 4-8 (b), one can observe that the resonance frequency of the middle beam does not shift with the distance as much as the 2-beam alignment. A cross section of the 3D plot on the highest voltage of the middle beam, where the distance between the beams is 0.072m, is plotted on Figure 4-8 (d). From this plot, the depression of the middle beam output after installing the stopper can be observed. Two peaks for each beam were created, one up leveraged by the new natural frequency of the upper beam, and the other frequency slightly raised from the middle beam. On the other hand, the side beam is slightly enhanced, and also stricken by the resonance frequency of the middle beam. The middle beam, with is designed with higher displacement, and thus it also pushes the two side beams to a higher displacement level comparing to their own resonance.

Dirac Response

The Dirac response was also simulated, by providing a pulse force into the system, the ring down of the middle beam was observed, with or without the top/bottom beams. Results in Figure 4-9 showed that the ring amplitude was increased, and the ring time was prolonged. By integrating the power with time within the first 10 seconds, the single beam has an output of 1.0907 nJ. While installed in the structure, the output is 1.43 nJ, which is a 1.31 fold enhance.

Figure 4-9 Simulation result of the Dirac response

4.2.2.2 The Asymmetric Alignment

When three beams are set with different resonance frequency or distance, it is possible for us to design an output curve with three peaks. Thereby, the variables can be the resonance frequency and also the distance. To slightly tune the resonance in experiment, one can alter the resonance frequency by simply adding additional proof masses on the tip of the beam. Another method is to relocate the proof masses.

Firstly, to keep the simulation simple, the distances of D12 and D13 are kept identical, whereas the proof mass is adjusted to tune the resonance frequency. As an example simulation and to distribute the resonance frequency, the M value of each beam are: 21, 15, and 4 g, based on the spring steel beam structure of the middle beam, which is 75x15x0.5 mm in length, width and thickness. The simulation results are shown in Figure 4-10. On the left, the connected beam output are shown, whilst on right the Figure 4-10 (a) shows the 3D plot showing the resonance shift and amplitude change corresponding to the different inter-beam distances. From Figure 4-10 (b) one can have a bird view of the frequency shift.

For a clear vision of the effect in the middle beam, it is cross-sectioned in D12=0.02 m, and shown in Figure 4-10 (c). The “unconnected sum” curve indicates the voltage summation of the beams when they are not magnetically connected. M-Connected Sum, on the other hand, is the sum of the voltage output from the magnetically connected beams. The first resonance has a strong influence on all three beams, while the second, dominated by beam 2, is observed weaker. From which, one can observe that the middle beam (Beam 1, loaded with 21 g proof mass) is subjected to both side beams (Beam 2, proof mass 15 g, and Beam 3, loaded with 4 g), but the influenced output was relatively low compared to resonance 1. The third resonance was dominated by the beam 3.

Observing the output, beam 2 and beam 3, which are smaller in mass, is unable to provide a higher influence comparing to beam 1.

Operating with beam 1 with a small mass and beam 2 and 3 with a higher could solve the problem.

However, beam 1 would own a higher frequency comparing to the other 2, and much higher when sandwiched between the two beams. The operating frequencies would be too far to be considered.

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Figure 4-10 Simulation results for asymmetrically aligned 3 beam array

(a) 3D plot of frequency – distance –voltage where m1=21, m2=15, m3=4, g=0.1g, and identical beam lengths (b) Bird view showing the frequency shift (c) cross section of D12=D13=0.02m

To avoid the problem mentioned, and to create an evenly distributed output, another configuration was designed. The beams have their proof mass kept identical. As shown in Figure 4-11,

the proof mass of beam 1 was moved to the middle of the beam, keeping identical beam length. In this way, beam 2 and 3 has enlarged force exerted to beam 1, which is considered shortened. On the other hand, beam 1 has its force reduced. Simulation results are shown in Figure 4-12. Figure 4-12 (a) shows the 3D plot of the three beams, connected and operated alone. From the right of Figure 4-12 (b), one can observe that the resonance frequency of beam 1 shifted, and became higher than beam 2 when they are operated separately. This enables beam 2 to dominate the first resonance, and beam 1 the second. From the cross section plot shown in Figure 4-12 (c), one can observe that the amplitude of each resonance is more evenly distributed.

Figure 4-11 A new configuration for 3-beam alignments

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Figure 4-12 Simulation results for the modified beam 1

(a) 3D plot of frequency – distance –voltage where m1=21, m2=15, m3=4, g=0.1g, m1 placed 25 mm from the tip and displacement =D12=D13 (b) Bird view showing the frequency shift (c) cross section

of D12=D13=0.0262m

To shift the location of different peaks, in our example, we could also fix D13, for example on D13=0.035m, and move only D12. In this way, one can define the operating bandwidth, as shown in Figure 4-13. The first and third resonance is relatively stable comparing to the shifting second resonance.

Figure 4-13 Bird view of frequency – distance –voltage for one fixed distance

where m1=21, m2=15, m3=4, g=0.1g showing the frequency shift when D13 is fixed, m1 placed 25 mm from the tip.

To bring the resonance closer, with m1 =14 g, m2=10 g, m3=4g , the cross section on D12

=D13=35 mm is shown as Figure 4-14. The overall output is increased, while the bandwidth is also increased.

Figure 4-14 Cross section of D12=D13=0.035 m;

m1=14, m2=10, m3=4, m1 placed 35 mm from tip. Driven on 0.1g.